How should I go about reading mathematics papers and textbooks as a PhD student

Question

It shames me to admit it, but I feel like I still haven’t figured out the right way to efficiently read papers and textbooks. I am a second year PhD student in pure mathematics, and I struggle a lot to balance the need to learn a lot new mathematics with the fact that I have a finite amount of time in which to do it.

I should have asked this question years ago: what is best (i.e. most efficient) way to read papers and textbooks in order to have a working knowledge of the subject?

Maybe I should be more precise. I am working in both algebraic geometry and homotopy theory (think simplicial presheaves and K-Theory), and I have recently found myself overwhelmed by the amount that I need to learn. In an ideal world I would read background material by doing every exercise in every textbook, and by working carefully through every proof in every paper, but I worry that I just don’t have enough time. On the flip side, I often find myself “reading” mathematics without actually absorbing any working knowledge, so I basically don’t make any direct research progress.

I understand that reading in great volume is still constructive, and I have certainly learnt a lot about how mathematics fits together, but when I actually need to do new mathematics I consistently find myself lost.

To rephrase my question: can anyone offer an advice on their workflow when it comes to learning new mathematics?

Perhaps the way I feel is more or less how everyone feels, and it is confidence and organisation which is the problem. If that is the case, then I ask: can anyone offer advice on how to organise oneself day to day in a PhD to be productive? Moreover, can anyone offer advice on how to break out of a lack of self confidence when it comes to doing research mathematics?

I apologise in advance if this question has been asked many times before, but I haven’t been able to find the right thread. I apologise also if my questions are too multi-pronged, I just feel that they are all to interconnected to be split up.

Answer 1

I struggle a lot to balance the need to learn a lot new mathematics with the fact that I have a finite amount of time in which to do it.

This is something that almost everyone struggles with when they're starting out, and it is one of the reasons having a good advisor is important: to tell you what specifically you should be working on.

Roughly, the point undergraduate and Master's programs or the first year or two of a PhD program in the US is to give you a solid base in mathematics. But the main point of (the PhD part of) a PhD program is to train you to be able to do research. It is not meant to give you a comprehensive understanding of your area of interest (which in most cases is impossible anyway).

Ideally, over the course of a PhD, you learn enough of the general context to be able:

1. to formulate interesting questions,

2. determine (more or less) what is known about these questions in the literature;

and learn enough relevant technical abilities to:

3. make new progress on or solve specific problems.

Typically once you get to a point in the general context of having specific problems to work on, you should spend more time (maybe 80% of your research time?) on your specific problems than on trying understand the "big picture." As I also quoted in this answer, Ravi Vakil says

...mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards".

As for specific advice:

1. Discuss this with your advisor. See if they have specific suggestions for things you should read, both for general understanding and for your specific problem.

2. Try to organize different tasks with a schedule. For instance, maybe work on your thesis problem 4 days a week, and read some general literature one day a week. Or you can split things up by times of day or weekdays/weekends. Some people work better with a strict schedule, I prefer a more flexible schedule. Maybe if I'm feeling stuck on my research one day, I'll do other tasks that day. Or one day something makes me curious about another topic, I learn about that.

3. Participate in activities to learn things outside of your thesis problem: seminars, conferences, workshops, summer schools, etc. If there's something you want to learn in depth, maybe organize a seminar with other students/postdocs.

4. To learn things a bit outside your thesis focus, you can also try reading survey papers, books, lecture notes and introductions of research papers. Also talk to people. When looking at papers, one often just reads the introduction and maybe skims relevant bits to see what the paper is about. Normally it is only if you need to know details for something you are doing that read the paper more carefully (and then sometimes just one specific section). One thing to help you absorb what the paper is about is to think about what the results say in special cases or how it compares with other results you know. If you're doing more than skimming, it's also helpful to "read actively" by writing out what is going on, together with maybe some examples, in a notebook as you are reading.

5. If you are going through books/notes with exercises, it is up to you how much time you want to spend on exercises. As long as you stick to your schedule, doing exercises won't take time away from your thesis problem, only slow down the rate at which you're going through other material.

6. Writing things down helps force you to understand them, and is useful to organize your thoughts when there's a lot of material. Try typing up notes for yourself about the topic you are learning. You can summarize the main results and ideas, work out various examples, etc.

7. Be patient. There is a lot to learn, and it takes time. Gradually you'll absorb more and more, and you do need to make an effort, but a lot of it will come naturally from working on specific problems.

Also, re: the lack of self-confidence, check out How should I deal with discouragement as a graduate student? and maybe some other threads on the impostor syndrome.

Answer 2

If you already have a good sense of how the relevant fields fit together, and you know your way around the standard references, I personally find that the best approach is to just jump into a good paper. Really studying a paper related to your research is a much better way to learn specific techniques than grinding through textbook exercises. I think there's generally too much emphasis on "background" at the expense of working on new mathematics.

Answer 3

I'm a second year student too.

I think that one have to study what really needs to know, for instance, if I need the Feit-Thompson Theorem (for some reason) I cite and use, if you need to use the tools and ideas that were used in that long proof, then you read the paper. Now, how to read it, in first instance, keep the central ideas and go deeper if you really need it for your work. There is a lot of mathematics, and nobody can learn everything, even if you only study algebraic geometry (it is a broad field). My tutor says that one will have a lot of years to study and go deep in topics that are of our interest, and in the moment, work hard for getting the PhD (it is not easy, we know it), and publish